Method for fault diagnosis of aero-engine sensor and actuator based on LFT

ABSTRACT

The present invention discloses a method for fault diagnosis of the sensors and actuators of an aero-engine based on LFT, and belongs to the field of fault diagnosis of aero-engines. The method comprises: establishing an aero-engine state space model using a combination of a small perturbation method and a linear fitting method; establishing an affine parameter-dependent linear-parameter-varying (LPV) model of the aero-engine based on the model; converting the LPV model of the aero-engine having perturbation signals and sensor and actuator fault signals into a linear fractional transformation (LFT) structure to obtain an synthesis framework of an LPV fault estimator; solving a set of linear matrix inequalities (LMIs) to obtain the solution conditions of the fault estimator; and designing the fault estimator in combination with the LFT structure to realize fault diagnosis of the sensors and actuators of an aero-engine.

TECHNICAL FIELD

The present invention belongs to the field of fault diagnosis ofaero-engines, and particularly relates to a method for fault diagnosisof an aero-engine sensor and actuator based on LFT.

BACKGROUND

An aero-engine is an important component of an aircraft, and the healthof flight states directly depends on the health of the aero-engine. Asan underlying information collection platform of the aero-engine, anaero-engine sensor can accurately measure the state information of thecomponent and the system in the operation process of the aero-engine, soas to construct an effective control system. If the aero-engine sensorfails, accurate performance and state parameters cannot be provided forthe control system, and accurate control cannot be achieved. Anaero-engine actuator is an important link for connecting the aero-engineand the control system. The actuator makes the controlled object changeaccording to a control command given by the control system, so as tocontrol the operation state of the aero-engine. If the aero-engineactuator fails, wrong information will be provided for the controlsystem, which will bring hidden danger to the safety of the aircraft andmay cause catastrophic consequences. Therefore, the present inventionmonitors the performance of the sensors and the actuators of theaero-engine, and has an important significance for real-time diagnosisand alarm of a fault state. Literatures have shown that, firstly, theexisting technology of fault diagnosis of the aero-engine sensor andactuator mainly focuses on fault detection, that is, the existingtechnology can only decide whether the aero-engine sensor and theactuator fail, and few researches are made on the estimation method ofthe fault signal. Specifically, different fault modes of the sensor andthe actuator correspond to different fault treatment measures. For thefault of the aero-engine sensor, the actual fault is mainly drift. Ifthe sensor drifts, the sensor measurement information can be correctedthrough controller design. The faults of the aero-engine actuatorinclude degradation, drift, jamming, etc. If the actuator degrades ordrifts, the aero-engine can be normally operated through the controllerdesign; and if the actuator jams, the actuator needs to be switched to aredundancy mode, and the actuator is maintained after the flight iscompleted. Therefore, only the fault detection is researched, that is,whether the fault occurs, which is not conducive to the safe operationand maintenance of the aero-engine. Reliable estimation of the severityof fault can realize accurate judgment of the states of the aero-enginesensor and actuator and simultaneously reduce the maintenance cost ofthe aero-engine. In addition, the aero-engine can be described as atypical linear-parameter-varying (LPV) system. The literature shows thatthe fault estimation methods of the LPV system are mainly classifiedinto two categories in recent years: the first category of methods is anobserver-based method, but such methods are not robust to perturbationand model uncertainty in the system, that is, the errors caused byexternal perturbation and modeling of the system will seriously affectthe observation result of an observer. The other category is a faultestimation method based on H_(∞) optimization technology. The method canimprove the robustness of the system, but the fault estimation researchbased on this technology is still in the beginning stage, and stillneeds to further discuss many issues.

SUMMARY

In view of the problem that a fault signal cannot be accuratelyestimated under external disturbance and modeling errors in the existingtechnology of fault diagnosis of an aero-engine sensor and actuator, thepresent invention provides a method for fault diagnosis of the sensorand actuator of an aero-engine based on LFT, which can adaptively adjustthe parameters of a fault estimator according to the change of theparameters in an LPV model of the aero-engine, realize rapid detectionof faults in the system, accurately reconstruct the fault signal,propose maintenance recommendations in time and provide better basicguarantee for later fault-tolerant control.

To achieve the above purpose, the technical solution adopted by thepresent invention comprises the following steps:

A method for fault diagnosis of the sensor and actuator of anaero-engine based on LFT comprises the following steps:

step 1: establishing an aero-engine state space model using acombination of a small perturbation method and a linear fitting method;

step 2: establishing an affine parameter-dependentlinear-parameter-varying (LPV) model of the aero-engine;

step 3: converting the affine parameter-dependent LPV model of theaero-engine having perturbation and sensor and actuator fault into alinear fractional transformation (LFT) structure, and establishing anH₂₈ synthesis framework of an LPV fault estimator of the aero-engine;

step 4: solving a set of linear matrix inequalities (LMIs) to obtain thesolution conditions of the fault estimator;

step 5: designing the fault estimator in combination with the LFTstructure to realize fault diagnosis of the sensor and actuator of theaero-engine.

The step 1 comprises the following steps:

step 1.1: inputting fuel pressure p_(f) under steady operating pointsinto an aero-engine; inputting a fuel pressure step signal U_(pf1) ,with an amplitude of 0.01 p_(f) into the aero-engine after relativeconversion speed n_(h) of a high pressure rotor of the aero-enginereaches a corresponding steady state; and respectively collectingrelative conversion speed response Y_(nh1) of the high pressure rotorand relative conversion speed response Y_(nl1) of a low pressure rotoroutputted by the aero-engine;

step 1.2: repeating the process of the step 1 for N times; andrespectively collecting the relative conversion speed response Y_(nhi)of the high pressure rotor and the relative conversion speed responseY_(nli) of the low pressure rotor under given fuel pressure p_(fi),wherein i=1,2,3, . . . , N;

step 1.3: constructing a discrete small perturbation state space modelof the aero-engine under the steady operating points according to thelinear fitting method by taking the fuel pressure step signal U _(pfi)as an input variable and taking the relative conversion speed response)Y_(nhi) of the high pressure rotor and the relative conversion speedresponse Y_(nli) of the low pressure rotor as state variables;

step 1.4: converting the discrete small perturbation state space modelof the aero-engine under the steady operating points into a continuoussmall perturbation state space model according to a sampling period T toobtain the state space model of the aero-engine;

$\begin{matrix}\left\{ \begin{matrix}{{\overset{.}{x}}_{p} = {{A_{pi}x_{p}} + {B_{pi}u}}} \\{y_{p} = {{C_{pi}x_{p}} + {D_{pi}u}}}\end{matrix} \right. & (1)\end{matrix}$

wherein the state variable is x_(p)=[Y_(nl) ^(T) Y_(nh) ^(T)]^(T)∈R^(n);{dot over (x)}_(p) represents a first derivative of x_(p); an inputvariable is u=U_(pf)∈R^(t); an output variable is y_(p)=Y_(nh)∈R^(m);A_(pi), B_(pi), C_(pi) and D_(pi) are system state space matrices;C_(pi)=C_(p)=[0 1]; D_(pi)=D_(p)=0; R^(n), R^(t) and R^(m) respectivelyrepresent sets of real numbers with dimensions of n, t and m; Trepresents transposing for the matrices.

The step 2 comprises the following steps:

step 2.1: setting the relative conversion speed n_(hi) of the highpressure rotor of the aero-engine as a scheduling parameter θ(i),i=1,2,3, . . . , N;

step 2.2: expressing a system matrix A_(p)(θ) and a control matrixB_(p)(θ) of the continuous small perturbation state space model of theaero-engine as affine parameter-dependent forms, as follows:A _(p)(θ)=A ₀ +θA ₁, B _(p)(θ)=B ₀ +θB ₁  (2)

wherein A₀, A₁, B₀ and B₁ respectively represent coefficient matrices tobe solved; rewriting the formula (2) into

$\begin{matrix}{{{A_{p}(\theta)} = {\begin{bmatrix}I & {\theta\; I}\end{bmatrix}\begin{bmatrix}A_{0} \\A_{1}\end{bmatrix}}},\mspace{14mu}{{B_{p}(\theta)} = {\begin{bmatrix}I & {\theta\; I}\end{bmatrix}\begin{bmatrix}B_{0} \\B_{1}\end{bmatrix}}}} & (3)\end{matrix}$

wherein I is a unit matrix;

then

$\begin{matrix}{{\begin{bmatrix}A_{0} \\A_{1}\end{bmatrix} = {\begin{bmatrix}I & {\theta\; I}\end{bmatrix}^{+}{A_{p}(\theta)}}},\mspace{14mu}{\begin{bmatrix}B_{0} \\B_{1}\end{bmatrix} = {\begin{bmatrix}I & {\theta\; I}\end{bmatrix}^{+}{B_{p}(\theta)}}}} & (4)\end{matrix}$

wherein [I θI]⁺ is Moore-Penrose pseudo- inverse of [I θI], i.e., thesystem matrix A_(p)(θ) and the control matrix B_(p)(θ) of the solvedaffine parameter-dependent LPV model of the aero-engine;

step 2.3: establishing the affine parameter-dependent LPV model of theaero-engine{dot over (x)} _(p) =A _(p)(θ)x _(p) +B _(p)(θ)uy _(p) =C _(p) x _(p) +D _(p) u   (5).

The step 3 of establishing an H_(∞) synthesis framework of an LPV faultestimator of the aero-engine comprises the following steps:

step 3.1: expressing the affine parameter-dependent LPV model P(s, θ) ofthe aero-engine with perturbation and sensor and actuator fault intox _(p) =A _(p)(θ)x _(p) +B _(p)(θ)u+E _(p) d+F _(p) fy _(p) =C _(p) x _(p) +D _(p) u+G _(p) d+H _(p) f   (6)

wherein d∈R^(q) is a perturbation signal; f∈R^(l) is a fault signalcomprising sensor fault and actuator fault; R^(q) and R^(l) respectivelyrepresent sets of real numbers with dimensions of q and l; E_(p), F_(p),G_(p) and H_(p) are system state space matrices; an upper LFT structureof P (s, θ) is expressed into

$\begin{matrix}\left\{ \begin{matrix}{{\begin{bmatrix}{\overset{.}{x}}_{p} \\z_{\theta} \\y_{p}\end{bmatrix} = {\begin{bmatrix}A_{p} & B_{p\;\theta} & B_{pw} \\C_{p\;\theta} & {D_{p}\theta\;\theta} & D_{p\;\theta\; w} \\C_{pw} & D_{{pw}\;\theta} & D_{pww}\end{bmatrix}\begin{bmatrix}x_{p} \\w_{\theta} \\w\end{bmatrix}}},{w = \begin{bmatrix}u \\d \\f\end{bmatrix}}} \\{w_{\theta} = {{\Delta(\theta)}z_{\theta}}}\end{matrix} \right. & (7)\end{matrix}$

wherein an external input variable is w=[u^(T) d^(T) f^(T)]^(T)∈R^(p1);w_(θ)∈R^(r) is an output variable of a time varying part Δ(θ)=θI;z_(θ)∈R^(r) is an input variable of the time varying part Δ(θ)=θI;A_(p), B_(pθ), B_(pw), C_(pθ), C_(pw), D_(pθθ), D_(pθw), D_(pwθ) andD_(pww) are system state space matrices; R ^(p1) and R^(r) respectivelyrepresent sets of real numbers with dimensions of p1 and r; p1=t+q+l,i.e., the dimension p1 of the external input variable w is equal to thesum of the dimension t of the input variable u of the aero-engine, thedimension q of the perturbation signal d and the dimension l of thefault signal f;

step 3.2: setting the form of the fault estimator K(s,θ) as follows

$\begin{matrix}\left\{ \begin{matrix}{{\overset{.}{x}}_{K} = {{{A_{K}(\theta)}x_{K}} + {{B_{K}(\theta)}u_{K}}}} \\{\overset{\hat{}}{f} = {{{C_{K}(\theta)}x_{K}} + {{D_{K}(\theta)}u_{K}}}}\end{matrix} \right. & (8)\end{matrix}$

wherein x_(k)∈R^(k) is a state variable of the fault estimator K(s,θ);{dot over (x)}_(k) represents a first derivative of x_(K); R^(k)represents a set of real numbers with a dimension of k; u_(K)=[u^(T)y_(p) ^(T)]^(t)∈R^(p2) is an input variable of K(s,θ); p2=t+m, i.e., thedimension p2 of the input variable u_(K) of K (s,θ) is equal to the sumof the dimension t of the input variable u of the aero-engine and thedimension m of the output variable y_(p) of the aero-engine; {circumflexover (f)}∈R^(l) is an output variable of K(s,θ), i.e., an estimatedvalue of the fault signal f; A_(K)(θ), B_(K)(θ), C_(K)(θ) and D_(K)(θ)are system state space matrices; K(s,θ) is express into a lower LFTstructure as follows:

$\begin{matrix}\left\{ \begin{matrix}{{\begin{bmatrix}{\overset{.}{x}}_{K} \\\hat{f} \\z_{K}\end{bmatrix} = {\begin{bmatrix}A_{K} & B_{K\; 1} & B_{K\;\theta} \\C_{K\; 1} & D_{K\; 11} & D_{K\; 1\;\theta} \\C_{K\;\theta} & D_{K\;\theta\; 1} & D_{K\;{\theta\theta}}\end{bmatrix}\begin{bmatrix}x_{K} \\u_{K} \\w_{K}\end{bmatrix}}},{u_{K} = \begin{bmatrix}u \\y_{p}\end{bmatrix}}} \\{w_{k} = {{\Delta_{k}(\theta)}z_{k}}}\end{matrix} \right. & (9)\end{matrix}$

wherein w_(K)∈R^(r) is an output variable of the time varying partΔ_(K)(θ)=θI; z_(K)∈R^(r) is an input variable of the time varying partΔ_(k)(θ)=θI; A_(K), B_(K1), B_(Kθ), C_(Kθ), C_(K1), C_(Kθ), D_(K11),D_(K1θ), D_(Kθ1) and D_(Kθθ) are system state space matrices;

step 3.3: according to the time varying part Δ(θ) in the LPV modelP(s,θ) of the aero-engine and the time varying part Δ_(K)(θ) in thefault estimator K(s,θ), expressing the H_(∞) synthesis framework of theLPV fault estimator as:

$\begin{matrix}{\begin{bmatrix}{\overset{.}{x}}_{p} \\\frac{{\overset{.}{x}}_{K}}{z_{K}} \\\frac{z_{\theta}}{e_{f}}\end{bmatrix} = {\begin{bmatrix}\overset{\_\;}{A} & {\overset{\_\;}{B}}_{\theta} & {\overset{\_\;}{B}}_{w} \\{\overset{\;\_}{C}}_{\theta} & {\overset{\;\_}{D}}_{\theta\theta} & {\overset{\;\_}{D}}_{\theta w} \\{\overset{\;\_}{C}}_{w} & {\overset{\;\_}{D}}_{w\;\theta} & {\overset{\;\_}{D}}_{ww}\end{bmatrix}\begin{bmatrix}x_{p} \\\frac{x_{K}}{w_{K}} \\\frac{w_{\theta}}{w}\end{bmatrix}}} & (10)\end{matrix}$

wherein e_(f)={circumflex over (f)}−f is a fault estimation error;system matrix Ā=A₀+T₁ΩT₂; system matrix B _(θ)=B₀₁+T₁ΩT₃; system matrixB _(w)=B₀₂+T₁ΩT₄; system matrix C _(θ)=C₀₁+T₅ΩT₂; system matrix D_(θθ)=D₀₁+T₅ΩT₃; system matrix D _(θw)=D₀₂+T₅ΩT₄; system matrix C_(w)=C₀₂+T₆ΩT₂; system matrix D _(wθ)=D₀₃+T₆ΩT₃; system matrix D_(ww)=D₀₄+T₆ΩT₄; fault estimator matrix

${\Omega = \begin{bmatrix}A_{K} & B_{K\; 1} & B_{K\;\theta} \\C_{K\; 1} & D_{K\; 11} & D_{K\; 1\;\theta} \\C_{K\;\theta} & D_{K\;\theta\; 1} & D_{K\;{\theta\theta}}\end{bmatrix}};$matrix

${T_{1} = \begin{bmatrix}0 & B_{2} & 0_{n \times r} \\I_{k} & 0 & 0\end{bmatrix}};$matrix

${T_{2} = \begin{bmatrix}0 & I_{k} \\C_{2} & 0 \\0_{r \times n} & 0\end{bmatrix}};$matrix

${T_{3} = \begin{bmatrix}0_{k \times r} & 0 \\0 & D_{2\;\theta} \\I_{r} & 0\end{bmatrix}};$matrix

${T_{4} = \begin{bmatrix}0_{k \times p\; 1} \\D_{21} \\0_{r \times p\; 1}\end{bmatrix}};$matrix

${T_{5} = \begin{bmatrix}0_{r \times k} & 0 & I_{r} \\0 & D_{\theta\; 2} & 0\end{bmatrix}};$matrix T₆=[0_(p1×k) D₁₂ 0_(p1×r)]; matrix

${A_{0} = \begin{bmatrix}A & 0 \\0 & 0_{k}\end{bmatrix}};$matrix

${B_{01} = \begin{bmatrix}0 & B_{\theta} \\0_{k \times r} & 0\end{bmatrix}};$matrix

${B_{02} = \begin{bmatrix}B_{1} \\0_{k \times p\; 1}\end{bmatrix}};$matrix

${C_{01} = \begin{bmatrix}0 & 0_{r \times k} \\C_{\theta} & 0\end{bmatrix}};$matrix

${D_{01} = \begin{bmatrix}0_{r} & 0 \\0 & D_{\theta\theta}\end{bmatrix}};$matrix

${D_{02} = \begin{bmatrix}0_{r \times p\; 1} \\D_{\theta\; 1}\end{bmatrix}};$matrix C₀₂=[C₁ 0_(p1×k)]; matrix D₀₃=[0_(p1×r) D_(1θ)]; matrix D₀₄=D₁₁;matrix A=A_(p); matrix B_(θ)=B_(pθ); matrix B₁=B_(pw); matrixB₂=0_(n×l); matrix C_(θ)=C_(pθ); matrix D_(θθ)=D_(pθθ); matrixD_(θ1)=D_(pθw); matrix D_(θ2)=0_(r×l); matrix C₁=0_(p1×n); matrixD_(1θ)=0_(p1×r); matrix

${D_{11} = \begin{bmatrix}0_{l \times {({{p\; 1} - l})}} & {- I_{l}} \\0_{{({{p\; 1} - l})} \times {({{p\; 1} - l})}} & 0_{{({{p\; 1} - l})} \times l}\end{bmatrix}};$matrix

${D_{12} = \begin{bmatrix}I_{l} \\0_{{({{p\; 1} - l})} \times l}\end{bmatrix}};$matrix

${C_{2} = \begin{bmatrix}0_{t \times n} \\C_{pw}\end{bmatrix}};$matrix

${D_{2\;\theta} = \begin{bmatrix}0_{t \times r} \\D_{{pw}\;\theta}\end{bmatrix}};$matrix

${D_{21} = \begin{bmatrix}I_{t} & 0_{t \times q} & 0_{t \times l} \\\; & D_{pww} & \;\end{bmatrix}};$matrix D₂₂=0_(p2×l); n represents the dimension of the state variablex_(p) of the aero-engine; r represents the dimension of the outputvariable w_(θ) of the time varying part Δ(θ) and the output variablew_(K) of the time varying part Δ_(K)(θ); k represents the dimension ofthe state variable x_(K) of the fault estimator K(s,θ).

The step 4 of obtaining the solution conditions of the fault estimatorcomprises the following steps:

step 4.1: obtaining the solution conditions of the fault estimatorK(s,θ), i.e.,

$\begin{matrix}{{{{\begin{bmatrix}I & 0 & 0 \\0 & I & 0 \\0 & 0 & I \\{\overset{\_}{A}\;} & {\overset{\_}{B}\;}_{\theta} & {\overset{\_}{B}\;}_{w} \\{\overset{\_}{C}\;}_{\theta} & {\overset{\_}{D}\;}_{\theta\theta} & {\overset{\_}{D}\;}_{\theta\; w} \\{\overset{\_}{C}\;}_{w} & {\overset{\_}{D}\;}_{w\;\theta} & {\overset{\_}{D}\;}_{ww}\end{bmatrix}^{T}\left\lbrack \begin{matrix}0 & 0 & 0 & X & 0 & 0 \\0 & Q & 0 & 0 & S & 0 \\0 & 0 & {{- \gamma}\; I} & 0 & 0 & 0 \\X & 0 & 0 & 0 & 0 & 0 \\0 & S^{T} & 0 & 0 & R & 0 \\0 & 0 & 0 & 0 & 0 & {\frac{1}{\gamma}I}\end{matrix} \right\rbrack}\left\lbrack \begin{matrix}I & 0 & 0 \\0 & I & 0 \\0 & 0 & I \\{\overset{\_}{A}\;} & {\overset{\_}{B}\;}_{\theta} & {\overset{\_}{B}\;}_{w} \\{\overset{\_}{C}\;}_{\theta} & {\overset{\_}{D}\;}_{\theta\theta} & {\overset{\_}{D}\;}_{\theta\; w} \\{\overset{\_}{C}\;}_{w} & {\overset{\_}{D}\;}_{w\;\theta} & {\overset{\_}{D}\;}_{ww}\end{matrix} \right\rbrack} < 0}\;} & (11) \\{\mspace{79mu}{{\begin{bmatrix}{\Delta(\theta)} & 0 \\0 & {\Delta_{K}(\theta)} \\I & 0 \\0 & I\end{bmatrix}^{T}{P\begin{bmatrix}{\Delta(\theta)} & 0 \\0 & {\Delta_{K}(\theta)} \\I & 0 \\0 & I\end{bmatrix}}} > 0}} & (12)\end{matrix}$

wherein X is a symmetric positive-definite matrix; a full block scalingmatrix

$P = \begin{bmatrix}Q & S \\S^{T} & R\end{bmatrix}$is a symmetric matrix; γ>0 is a performance index; Q, S and Rrespectively represent subscalar block matrices of P;

step 4.2: partitioning the symmetric positive-definite matrix X and aninverse matrix X⁻¹ thereof;

$\begin{matrix}{{X = \begin{bmatrix}L & M \\M^{T} & E\end{bmatrix}},{X^{- 1} = \begin{bmatrix}J & N \\N^{T} & F\end{bmatrix}}} & (13)\end{matrix}$

wherein L, M and E respectively represent block matrices of X; J, N andF respectively represent sub-block matrices of X⁻¹;

partitioning the full block scaling matrix P and the inverse matrix{tilde over (P)} thereof

$\begin{matrix}{{P = {\begin{bmatrix}Q & S \\S^{T} & R\end{bmatrix} = \begin{bmatrix}Q_{1}^{\;} & Q_{2}^{\;} & S_{1}^{\;} & S_{2}^{\;} \\Q_{2}^{T} & Q_{3}^{\;} & S_{3}^{\;} & S_{4}^{\;} \\S_{1}^{T} & S_{3}^{T} & R_{1} & R_{2} \\S_{2}^{T} & S_{4}^{T} & R_{2}^{T} & R_{3}\end{bmatrix}}},{\overset{\sim}{P} = {\begin{bmatrix}\overset{\sim}{Q} & \overset{\sim}{S} \\{\overset{\sim}{S}}^{T} & \overset{\sim}{R}\end{bmatrix} = \begin{bmatrix}{\overset{\sim}{Q}}_{1}^{\;} & {\overset{\sim}{Q}}_{2}^{\;} & {\overset{\sim}{S}}_{1}^{\;} & {\overset{\sim}{S}}_{2}^{\;} \\{\overset{\sim}{Q}}_{2}^{T} & {\overset{\sim}{Q}}_{3}^{\;} & {\overset{\sim}{S}}_{3}^{\;} & {\overset{\sim}{S}}_{4}^{\;} \\{\overset{\sim}{S}}_{1}^{T} & {\overset{\sim}{S}}_{3}^{T} & {\overset{\sim}{R}}_{1} & {\overset{\sim}{R}}_{2} \\{\overset{\sim}{S}}_{2}^{T} & {\overset{\sim}{S}}_{4}^{T} & {\overset{\sim}{R}}_{2}^{T} & {\overset{\sim}{R}}_{3}\end{bmatrix}}}} & (14)\end{matrix}$

wherein Q₁, Q₂ and Q₃ respectively represent sub-block matrices of Q;S₁, S₂, S₃ and S₄ respectively represent sub-block matrices of S; R₁, R₂and R₃ respectively represent sub-block matrices of R; {tilde over (Q)},{tilde over (S)} and {tilde over (R)} respectively represent sub-blockmatrices of {tilde over (P)}; {tilde over (Q)}₁, {tilde over (Q)}₂ and{tilde over (Q)}₃ respectively represent sub-block matrices of {tildeover (Q)}; {tilde over (S)}₁, {tilde over (S)}₂, {tilde over (S)}₃ and{tilde over (S)}₄ respectively represent sub-block matrices of {tildeover (S)}; {tilde over (R)}₁, {tilde over (R)}₂ and {tilde over (R)}₃respectively represent sub-block matrices of {tilde over (R)};

simplifying the solution conditions of the fault estimator K(s,θ), i.e.,

$\begin{matrix}{{{{{N_{L}^{T}\begin{bmatrix}I & 0 & 0 \\0 & I & 0 \\0 & 0 & I \\A & B_{\theta} & B_{1} \\C_{\theta} & D_{\theta\theta} & D_{\theta\; 1} \\C_{1} & D_{1\;\theta} & D_{11}\end{bmatrix}}^{T}\begin{bmatrix}0 & 0 & 0 & L & 0 & 0 \\0 & Q_{3} & 0 & 0 & S_{4} & 0 \\0 & 0 & {{- \gamma}\; I} & 0 & 0 & 0 \\L & 0 & 0 & 0 & 0 & 0 \\0 & S_{4}^{T} & 0 & 0 & R_{3} & 0 \\0 & 0 & 0 & 0 & 0 & {\frac{1}{\gamma}I}\end{bmatrix}}\begin{bmatrix}I & 0 & 0 \\0 & I & 0 \\0 & 0 & I \\A & B_{\theta} & B_{1} \\C_{\theta} & D_{\theta\theta} & D_{\theta\; 1} \\C_{1} & D_{1\;\theta} & D_{11}\end{bmatrix}}N_{L}} < 0} & (15) \\{{{{N_{J}^{T}\begin{bmatrix}{- A^{T}} & {- C_{\theta}^{T}} & {- C_{1}^{T}} \\{- B_{\theta}^{T}} & {- D_{\theta\theta}^{T}} & {- D_{1\;\theta}^{T}} \\{- B_{1}^{T}} & {- D_{\theta\; 1}^{T}} & {- D_{11}^{T}} \\I & 0 & 0 \\0 & I & 0 \\0 & 0 & I\end{bmatrix}}^{T}\quad}\left\lbrack \begin{matrix}0 & 0 & 0 & J & 0 & 0 \\0 & {\overset{\sim}{Q}}_{3} & 0 & 0 & {\overset{\sim}{S}}_{4} & 0 \\0 & 0 & {{- \frac{1}{\gamma}}I} & 0 & 0 & 0 \\J & 0 & 0 & 0 & 0 & 0 \\0 & {\overset{\sim}{S}}_{4}^{T} & 0 & 0 & {\overset{\sim}{R}}_{3} & 0 \\0 & 0 & 0 & 0 & 0 & {\gamma\; I}\end{matrix} \right\rbrack}{\quad{{\left\lbrack \begin{matrix}{- A^{T}} & {- C_{\theta}^{T}} & {- C_{1}^{T}} \\{- B_{\theta}^{T}} & {- D_{\theta\theta}^{T}} & {- D_{1\;\theta}^{T}} \\{- B_{1}^{T}} & {- D_{\theta\; 1}^{T}} & {- D_{11}^{T}} \\I & 0 & 0 \\0 & I & 0 \\0 & 0 & I\end{matrix} \right\rbrack N_{J}} > 0}}} & (16) \\{\mspace{79mu}{\begin{bmatrix}L & I \\I & J\end{bmatrix} > 0}} & (17) \\{\mspace{79mu}{{R > 0},{Q = {- R}},{{S + S^{T}} = 0}}} & (18)\end{matrix}$

wherein N_(L) and N_(J) respectively represent the bases of the nuclearspaces of [C₂ D_(2θ) D₂₁] and [B₂ ^(T) D_(θ2) ^(T) D₁₂ ^(T)];

step 4.3: solving the LMIs (15)-(18) to obtain matrix solutions L, J,Q₃, {tilde over (R)}₃, S₄ and {tilde over (S)}₄;

the step 5 of designing the fault estimator in combination with the LFTstructure comprises the following steps:

step 5.1: solving the symmetric positive-definite matrix X, the fullblock scaling matrix P and the inverse matrix {tilde over (P)} thereoffrom the formulas (13) and (14) according to the solved matrix solutionsL, J, Q₃, {tilde over (R)}₃, S₄ and {tilde over (S)}₄;

step 5.2: according to Schur complement Lemma, expressing the LMI (11)as

$\begin{matrix}{\begin{bmatrix}{{{\overset{\_}{A}}^{T}X} + {X\;\overset{\_}{A}}} & {{X\;{\overset{\_}{B}}_{\theta}} + {{\overset{\_}{C}}_{\theta}^{T}S^{T}}} & {X\;{\overset{\_}{B}}_{w}} & {\overset{\_}{C}\;}_{\theta}^{T} & {\overset{\_}{C}\;}_{w}^{T} \\{{{\overset{\_}{B}}_{\theta}^{T}X} + {S\;{\overset{\_}{C}}_{\theta}}} & {Q + {{\overset{\_}{D}}_{\theta\theta}^{T}S^{T}} + {S\;{\overset{\_}{D}}_{\theta\theta}}} & {S\;{\overset{\_}{D}}_{\theta\; w}} & {\overset{\_}{D}\;}_{\theta\theta}^{T} & {\overset{\_}{D}\;}_{w\;\theta}^{T} \\{{\overset{\_}{B}\;}_{w}^{T}X} & {{\overset{\_}{D}\;}_{\theta\; w}^{T}S^{T}} & {{- \gamma}\; I} & {\overset{\_}{D}\;}_{\theta\; w}^{T} & {\overset{\_}{D}\;}_{ww}^{T} \\{\overset{\_}{C}\;}_{\theta} & {\overset{\_}{D}\;}_{\theta\theta} & {\overset{\_}{D}\;}_{\theta\; w} & {- \overset{\sim}{R}} & 0 \\{\overset{\_}{C}\;}_{w} & {\overset{\_}{D}\;}_{w\;\theta} & {\overset{\_}{D}\;}_{ww} & 0 & {{- \gamma}\; I}\end{bmatrix} < 0} & (19)\end{matrix}$

solving the LMI (19) to obtain a fault estimator matrix Ω;

step 5.3: obtaining a state space matrix of the fault estimator K(s,θ)

$\begin{matrix}{\begin{bmatrix}{A_{K}(\theta)} & {B_{K}(\theta)} \\{C_{K}(\theta)} & {D_{K}(\theta)}\end{bmatrix} = {\left\lbrack \begin{matrix}A_{K} & B_{K\; 1} \\C_{K\; 1} & D_{K\; 11}\end{matrix} \right\rbrack + {\quad{\begin{bmatrix}B_{K\;\theta} \\D_{K\; 1\;\theta}\end{bmatrix}{\Delta_{K}(\theta)}{{\left( {I - {D_{K\;{\theta\theta}}{\Delta_{K}(\theta)}}} \right)^{- 1}\begin{bmatrix}C_{K\;\theta} & D_{K\;\theta\; 1}\end{bmatrix}}.}}}}} & (20)\end{matrix}$

The present invention has the beneficial effects: the method for faultdiagnosis of the sensor and actuator of an aero-engine designed by thepresent invention respectively converts the LPV model and the faultestimator of the aero-engine into the time varying part and the LFTstructure formed by the time varying part, wherein the time varying partis changed with the change of the time varying parameter vector. Thus,the fault estimator has a gain scheduling characteristic and can realizeaccurate estimation of the fault signal under the influence of uncertainconditions such as external perturbation and modeling errors, so as tofacilitate in understanding information about the type, the generationtime and the severity of the fault. In addition, the present inventionreduces the design conservativeness of the fault estimator through the Sprocess.

DESCRIPTION OF DRAWINGS

FIG. 1 is a contrast curve of relative conversion speed response Y_(nh)of a high pressure rotor of a state space model of an aero-engine andtest data under H=0,Ma=0,n₂=90% operating state.

FIG. 2 is a contrast curve of relative conversion speed response Y_(nh)of a high pressure rotor of an LPV model of an aero-engine and test dataunder H=0,Ma=0,n₂=90% operating state.

FIG. 3 is a structural diagram of an upper LFT of an LPV model P(s,θ) ofan aero-engine.

FIG. 4 is a system structural diagram under an LFT framework.

FIG. 5 is an H_(∞) synthesis framework of an LPV fault estimator.

FIG. 6(a) and FIG. 6(b) are simulation results of sudden faultestimation.

FIG. 7(a) and FIG. 7(b) are simulation results of slow fault estimation.

FIG. 8(a) and FIG. 8(b) are simulation results of intermittent faultestimation.

FIG. 9 is a flow chart of the present invention.

DETAILED DESCRIPTION

The embodiments of the present invention will be further described indetail below in combination with the drawings and the technicalsolution.

The flow chart of the present invention is shown in FIG. 9, andcomprises the following specific steps:

step 1.1: inputting fuel pressure p_(f) under steady operating pointsinto an aero-engine; inputting a fuel pressure step signal U_(pf1) withan amplitude of 0.01 p_(f) into the aero-engine after relativeconversion speed n_(h) of a high pressure rotor of the aero-enginereaches a corresponding steady state; and respectively collectingrelative conversion speed response Y_(nh1) of the high pressure rotorand relative conversion speed response Y_(nl1) of a low pressure rotoroutputted by the aero-engine.

Step 1.2: repeating the above process for 13 times, i.e., respectivelycollecting the relative conversion speed response Y_(nhi) of the highpressure rotor and the relative conversion speed response Y_(nli) of thelow pressure rotor under given fuel pressure p_(fi) at balance points of13 working conditions of (H=0,Ma=0,n_(h)=88%, 89%, . . . , 100%),wherein i=1,2,3, . . . ,13.

Step 1.3: by taking the fuel pressure step signal U_(pfi) as an inputvariable and taking the relative conversion speed response Y_(nhi) ofthe high pressure rotor and the relative conversion speed responseY_(nli) of the low pressure rotor as state variables, expressing adiscrete small perturbation state space model of the aero-engine as

$\begin{matrix}\left\{ \begin{matrix}{x_{p_{k + 1}} = {{E_{i}x_{p_{k}}} + {F_{i}u_{k}}}} \\{y_{p_{k}} = {{G_{i}x_{p_{k}}} + {H_{i}u_{k}}}}\end{matrix} \right. & (21)\end{matrix}$

wherein the state variable x_(p)=[Y_(nl) Y_(nh)]^(T)∈R^(n); the inputvariable u=U_(pf)∈R^(t); the output variable y_(p)=Y_(nh)∈R^(m),i=1,2,3, . . . ,13; the subscript k, k+1 are corresponding samplingmoments; E_(i), F_(i), G_(i) and H_(i) are system state space matriceswith appropriate dimensions; R^(n), R^(t) and R^(m) respectivelyrepresent sets of real numbers with dimensions of n, t and m; Trepresents transposing for the matrices. According to the basic idea ofthe fitting method, a linear least square problem is established forformula (21), and the system matrices E_(i), F_(i), G_(i), H_(i) aresolved by using the lsqnonlin function in MATLAB.

Step 1.4: converting the discrete small perturbation state space modelof the aero-engine under the steady operating points into a continuoussmall perturbation state space model according to a sampling period T=25ms to obtain the state space model of the aero-engine;

$\begin{matrix}\left\{ \begin{matrix}{{\overset{.}{x}}_{p} = {{A_{pi}x_{p}} + {B_{pi}u}}} \\{y_{p} = {{C_{pi}x_{p}} + {D_{pi}u}}}\end{matrix} \right. & (22)\end{matrix}$

wherein A_(pi), B_(pi), C_(pi) and D_(pi) are system state spacematrices with appropriate dimensions; C_(pi)=C_(p)=[0 1], andD_(pi)=D_(p)=0; and a relative conversion speed response Y_(nh) curve ofthe high pressure rotor of the state space model at the operating pintH=0,Ma=0,n₂=90% is provided, as shown in FIG. 1, and has an averagerelative error of 0.26% relative to the test data.

Step 2.1: setting the relative conversion speed n_(hi) of the highpressure rotor of the aero-engine as a scheduling parameterθ(i),i=1,2,3, . . . ,13.

Step 2.2: expressing a system matrix A_(p)(θ) and a control matrixB_(p)(θ) of the continuous small perturbation state space model of theaero-engine as affine parameter-dependent forms, as follows:A _(p)(θ)=A ₀ +θA ₁B _(p)(θ)=B ₀ +θB ₁   (23)

wherein A₀, A_(l), B₀ and B₁ respectively represent coefficient matricesto be solved.

Rewriting the formula (23) into

$\begin{matrix}{{{\begin{bmatrix}I & {{\theta(1)}I} \\I & {{\theta(2)}I} \\\vdots & \vdots \\I & {{\theta(13)}I}\end{bmatrix}\begin{bmatrix}A_{0} \\A_{1}\end{bmatrix}} = \begin{bmatrix}{A_{p}\left( {\theta(1)} \right)} \\{A_{p}\left( {\theta(2)} \right)} \\\vdots \\{A_{p}\left( {\theta(13)} \right)}\end{bmatrix}},{{\begin{bmatrix}I & {{\theta(1)}I} \\I & {{\theta(2)}I} \\\vdots & \vdots \\I & {{\theta(13)}I}\end{bmatrix}\begin{bmatrix}B_{0} \\B_{1}\end{bmatrix}} = \begin{bmatrix}{B_{p}\left( {\theta(1)} \right)} \\{B_{p}\left( {\theta(2)} \right)} \\\vdots \\{B_{p}\left( {\theta(13)} \right)}\end{bmatrix}}} & (24)\end{matrix}$

wherein I is a unit matrix.

Then

$\begin{matrix}{{\begin{bmatrix}A_{0} \\A_{1}\end{bmatrix} = {\begin{bmatrix}I & {{\theta(1)}I} \\I & {{\theta(2)}I} \\\vdots & \vdots \\I & {{\theta(13)}I}\end{bmatrix}^{+}\begin{bmatrix}{A_{p}\left( {\theta(1)} \right)} \\{A_{p}\left( {\theta(2)} \right)} \\\vdots \\{A_{p}\left( {\theta(13)} \right)}\end{bmatrix}}},{\begin{bmatrix}B_{0} \\B_{1}\end{bmatrix} = {\begin{bmatrix}I & {{\theta(1)}I} \\I & {{\theta(2)}I} \\\vdots & \vdots \\I & {{\theta(13)}I}\end{bmatrix}^{+}\begin{bmatrix}{B_{p}\left( {\theta(1)} \right)} \\{B_{p}\left( {\theta(2)} \right)} \\\vdots \\{B_{p}\left( {\theta(13)} \right)}\end{bmatrix}}}} & (25)\end{matrix}$

The pinv function in MATLAB is used to solve Moore-Penrose pseudo-inverse

${\begin{bmatrix}I & {{\theta(1)}I} \\I & {{\theta(2)}I} \\\vdots & \vdots \\I & {{\theta(13)}I}\end{bmatrix}^{+}\mspace{14mu}{{of}\mspace{14mu}\begin{bmatrix}I & {{\theta(1)}I} \\I & {{\theta(2)}I} \\\vdots & \vdots \\I & {{\theta(13)}I}\end{bmatrix}}},$and variable transformation is conducted on a variable parameter θ, sothat θ∈[−1,1], to obtain:

$\begin{matrix}{{A_{0} = \begin{bmatrix}{- 2.5839} & 0.5968 \\1.1613 & {- 4.3763}\end{bmatrix}},{A_{1} = \begin{bmatrix}0.4287 & {- 2.3152} \\0.3138 & {- 0.7456}\end{bmatrix}},{B_{0} = \begin{bmatrix}0.0037 \\0.0021\end{bmatrix}},{B_{1} = \begin{bmatrix}{- 0.0003} \\{- 0.0002}\end{bmatrix}}} & (26)\end{matrix}$

Step 2.3: establishing the affine parameter-dependent LPV model of theaero-engine{dot over (x)} _(p) =A _(p)(θ)x _(p) +B _(p)(θ)uy _(p) =C _(p) x _(p) +D _(p) u   (27)

wherein a relative conversion speed response Y_(nh) curve of the highpressure rotor of the LPV model of the aero-engine at the operating pintH=0,Ma=0,n₂=90% is provided, as shown in FIG. 2, and has an averagerelative error of 2.51% relative to the test data.

Step 3.1: expressing the affine parameter-dependent LPV model P(s,θ) ofthe aero-engine having perturbation and sensor and actuator fault intox _(p) =A _(p)(θ)x _(p) +B _(p)(θ)u+E _(p) d+F _(p) fy _(p) =C _(p) x _(p) +D _(p) u+G _(p) d+H _(p) f   (28)

wherein d∈R^(q) is a perturbation signal and takes Gaussian white noisewith standard deviation of 0.001; f∈R^(l) is a fault signal comprisingsensor fault and actuator fault, which respectively take sudden fault,slow fault and intermittent fault; R^(q) and R^(l) respectivelyrepresent sets of real numbers with dimensions of q and

${l;{E_{p} = \begin{bmatrix}0.01 \\0\end{bmatrix}}},{F_{p} = \begin{bmatrix}0 \\0.1\end{bmatrix}},$G_(p)=0.2, H_(p)=1.

The upper LFT structure of P (s,θ) can be expressed as the followingformula, as shown in FIG. 3:

$\begin{matrix}{y_{p} = {{F_{u}\left( {P^{\prime},{\Delta(\theta)}} \right)}\begin{bmatrix}u \\d \\f\end{bmatrix}}} & (29)\end{matrix}$

wherein F_(u) represents the upper LFT structure; P′ represents atime-invariable part in P(s,θ); Δ(θ)=θI represents a time varying partin P(s,θ), i.e.,

$\begin{matrix}\left\{ \begin{matrix}{{\begin{bmatrix}{\overset{.}{x}}_{p} \\z_{\theta} \\y_{p}\end{bmatrix} = {\begin{bmatrix}A_{p} & B_{p\;\theta} & B_{pw} \\C_{p\;\theta} & D_{p\;{\theta\theta}} & D_{p\;\theta\; w} \\C_{pw} & D_{{pw}\;\theta} & D_{pww}\end{bmatrix}\begin{bmatrix}x_{p} \\w_{\theta} \\w\end{bmatrix}}},{w = \begin{bmatrix}u \\d \\f\end{bmatrix}}} \\{w_{\theta} = {{\Delta(\theta)}z_{\theta}}}\end{matrix} \right. & (30)\end{matrix}$

wherein an external input variable is w=[u^(T) d^(T) f^(T)]^(T)∈R^(p1);w_(θ)∈R^(r) is an output variable of a time varying part Δ(θ)=θI;z_(θ∈R) ^(r) is an input variable of the time varying part Δ(θ)=θI;R^(p1) and R^(r) respectively represent sets of real numbers withdimensions of p1 and r; p1=t+q+l, i.e., the dimension p1 of the externalinput variable w is equal to the sum of the dimension t of the inputvariable u of the aero-engine, the dimension q of the perturbationsignal d and the dimension l of the fault signal f; the system statespace matrices are

$\begin{matrix}{\mspace{79mu}{{A_{p} = \begin{bmatrix}{- 2.5839} & 0.5968 \\1.1613 & {- 4.3763}\end{bmatrix}}{{B_{p} = \left\lbrack \begin{matrix}0.4287 & 0 & {- 2.3152} & 0 & {- 0.0003} & 0 \\0 & 0.3138 & 0 & {- 0.7456} & 0 & {- 0.0002}\end{matrix} \right\rbrack},\mspace{20mu}{B_{pw} = {\begin{bmatrix}0.0037 & 0.01 & 0 \\0.0021 & 0 & 1\end{bmatrix}\mspace{20mu}{C_{p\;\theta} = \begin{bmatrix}1 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 1 & 0 & 0\end{bmatrix}^{T}}}},{C_{pw} = {\begin{bmatrix}0 & 1\end{bmatrix}\mspace{20mu} D_{{p\;{\theta\theta}} = 0_{6 \times 6}}}},{D_{p\;\theta\; w} = \begin{bmatrix}0 & 0 & 0 & 0 & 1 & 1 \\0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0\end{bmatrix}^{T}},\mspace{20mu}{D_{{pw}\;\theta} = 0_{1 \times 6}},{D_{pww} = \begin{bmatrix}0 & 0.2 & 1\end{bmatrix}}}}} & (31)\end{matrix}$

Step 3.2: setting the form of the fault estimator K(s,θ) as follows

$\begin{matrix}\left\{ \begin{matrix}{{\overset{.}{x}}_{K} = {{{A_{K}(\theta)}x_{K}} + {{B_{K}(\theta)}u_{K}}}} \\{\overset{\hat{}}{f} = {{{C_{K}(\theta)}x_{K}} + {{D_{K}(\theta)}u_{K}}}}\end{matrix} \right. & (32)\end{matrix}$

wherein X_(K)∈R^(k) is a state variable of the fault estimator K (s,θ);u_(K)=[u^(T) y_(p) ^(T)]^(T)∈R^(p2) is an input variable of K(s,θ);p2=t+m, i.e., the dimension p2 of the input variable u_(K) of K (s,θ) isequal to the sum of the dimension t of the input variable u of theaero-engine and the dimension m of the output variable y_(p) of theaero-engine; {circumflex over (f)}∈R^(l) is an output variable of K(s,θ), i.e., the estimated value of the fault signal f; A_(K)(θ),B_(K)(θ), C_(K)(θ) and D_(K)(θ) are system state space matrices. K(s,θ)is expressed as a lower LFT structure as follows:

$\begin{matrix}{\overset{\hat{}}{f} = {{F_{l}\left( {K^{\prime},{\Delta_{K}(\theta)}} \right)}\begin{bmatrix}u \\y_{p}\end{bmatrix}}} & (33)\end{matrix}$

wherein F_(l) represents the lower LFT structure; K′ represents atime-invariable part in K(s,θ); Δ_(K)(θ)=θI represents a time varyingpart in K(s,θ), i.e.,

$\begin{matrix}\left\{ \begin{matrix}{{\begin{bmatrix}{\overset{.}{x}}_{K} \\\hat{f} \\z_{K}\end{bmatrix} = {\begin{bmatrix}A_{K} & B_{K\; 1} & B_{K\;\theta} \\C_{K\; 1} & D_{K\; 11} & D_{K\; 1\;\theta} \\C_{K\;\theta} & D_{K\;{\theta 1}} & D_{K\;{\theta\theta}}\end{bmatrix}\begin{bmatrix}x_{K} \\u_{K} \\w_{K}\end{bmatrix}}},{u_{k} = \begin{bmatrix}u \\y_{p}\end{bmatrix}}} \\{w_{K} = {{\Delta_{K}(\theta)}z_{K}}}\end{matrix} \right. & (34)\end{matrix}$

wherein w_(K)∈R^(r) is an output variable of the time varying partΔ_(K)(θ)=θI; z_(K)∈R^(r) is an input variable of the time varying partΔ_(K)(θ)=θI; A_(K), B_(K1), B_(Kθ), C_(K1), C_(Kθ), D_(K11), D_(K1θ),D_(Kθ1) and D_(Kθθ) are system state space matrices with appropriatedimensions.

Step 3.3: showing a system connection diagram under the LFT framework inFIG. 4, and giving a state space expression of the system P₁ in FIG. 4as

$\begin{matrix}{\begin{bmatrix}\overset{.}{x} \\z_{\theta} \\e_{f} \\u_{k}\end{bmatrix} = {\begin{bmatrix}A & B_{\theta} & B_{1} & B_{2} \\C_{\theta} & D_{\theta\theta} & D_{\theta\; 1} & D_{\theta\; 2} \\C_{1} & D_{1\;\theta} & D_{11} & D_{12} \\C_{2} & D_{2\;\theta} & D_{21} & D_{22}\end{bmatrix}\begin{bmatrix}x \\w_{\theta} \\w \\\hat{f}\end{bmatrix}}} & (35)\end{matrix}$

wherein system matrix A=A_(p); system matrix B_(θ)=B_(pθ); system matrixB₁=B_(pw); system matrix B₂=0_(n×l); system matrix C_(θ)=C_(pθ); systemmatrix D_(θθ)=D_(pθθ); system matrix D_(θ1)=D_(pθw); system matrixD_(θ2)=0_(r×l); system matrix C₁=0_(p1×n); system matrixD_(1θ)=0_(p1×r); system matrix

${D_{11} = \begin{bmatrix}0_{l \times {({{p\; 1} - l})}} & {- I_{l}} \\0_{{({{p\; 1} - l})} \times {({{p\; 1} - l})}} & 0_{{({{p\; 1} - l})} \times l}\end{bmatrix}};$system matrix

${D_{12} = \begin{bmatrix}I_{l} \\0_{{({{p\; 1} - l})} \times l}\end{bmatrix}};$system matrix

${C_{2} = \begin{bmatrix}0_{t \times n} \\C_{pw}\end{bmatrix}};$system matrix

${D_{2\;\theta} = \begin{bmatrix}0_{t \times r} \\D_{{pw}\;\theta}\end{bmatrix}};$system matrix

${D_{21} = \begin{bmatrix}I_{t} & 0_{t \times q} & 0_{t \times l} \\\; & D_{pww} & \;\end{bmatrix}};$D₂₂=0_(p2×l); n represents the dimension of the state variable x_(p) ofthe aero-engine; r represents the dimension of the output variable w_(θ)of the time varying part Δ(θ) and the output variable w_(K) of the timevarying part Δ_(K)(θ).

According to the time varying part Δ(θ) in the LPV model P(s,θ) of theaero-engine and the time varying part Δ_(K)(θ) in the fault estimator K(s,θ), expressing the H_(∞) synthesis framework of the LPV faultestimator as follows, as shown in FIG. 5

$\begin{matrix}{\begin{bmatrix}\overset{.}{x} \\\frac{{\overset{.}{x}}_{K}}{z_{K}} \\\frac{z_{\theta}}{e_{f}}\end{bmatrix} = {\begin{bmatrix}{\overset{\_}{A}\;} & {\overset{\_}{B}\;}_{\theta} & {\overset{\_}{B}\;}_{w} \\{\overset{\_}{C}\;}_{\theta} & {\overset{\_}{D}\;}_{\theta\theta} & {\overset{\_}{D}\;}_{\theta\; w} \\{\overset{\_}{C}\;}_{w} & {\overset{\_}{D}\;}_{w\;\theta} & {\overset{\_}{D}\;}_{ww}\end{bmatrix}\begin{bmatrix}x \\\frac{x_{K}}{w_{K}} \\\frac{w_{\theta}}{w}\end{bmatrix}}} & (36)\end{matrix}$

wherein e_(f)={circumflex over (f)}−f is a fault estimation error, i.e.,an output variable of the H_(∞) synthesis framework of the LPV faultestimator; system matrix Ā=A₀+T₁ΩT₂; system matrix B _(θ)=B₀₁+T₁ΩT₃;system matrix B _(w)=B₀₂+T₁ΩT₄; system matrix C _(θ)=C₀₁+T₅ΩT₂; systemmatrix D _(θθ)=D₀₁+T₅ΩT₃; system matrix D _(θw)=D₀₂+T₅ΩT₄; system matrixC _(w)=C₀₂+T₆ΩT₂; system matrix D _(wθ)=D₀₃+T₆ΩT₃; system matrix D_(ww)=D₀₄+T₆ΩT₄; fault estimator matrix

${\Omega = \begin{bmatrix}A_{K} & B_{K\; 1} & B_{K\;\theta} \\C_{K\; 1} & D_{K\; 11} & D_{K\; 1\;\theta} \\C_{K\;\theta} & D_{K\;\theta\; 1} & D_{K\;{\theta\theta}}\end{bmatrix}};$matrix

${T_{1} = \begin{bmatrix}0 & B_{2} & 0_{n \times r} \\I_{k} & 0 & 0\end{bmatrix}};$matrix

${T_{2} = \begin{bmatrix}0 & I_{k} \\C_{2} & 0 \\0_{r \times n} & 0\end{bmatrix}};$matrix

${T_{3} = \begin{bmatrix}0_{k \times r} & 0 \\0 & D_{2\;\theta} \\I_{r} & 0\end{bmatrix}};$matrix

${T_{4} = \begin{bmatrix}0_{k \times p\; 1} \\D_{21} \\0_{r \times p\; 1}\end{bmatrix}};$matrix

${T_{5} = \begin{bmatrix}0_{r \times k} & 0 & I_{r} \\0 & D_{\theta\; 2} & 0\end{bmatrix}};$matrix T₆=[0_(p1×k) D₁₂ 0_(p1×r)]; matrix

${A_{0} = \begin{bmatrix}A & 0 \\0 & 0_{k}\end{bmatrix}};$matrix

${B_{01} = \begin{bmatrix}0 & B_{\theta} \\0_{k \times r} & 0\end{bmatrix}};$matrix

${B_{02} = \begin{bmatrix}B_{1} \\0_{k \times p\; 1}\end{bmatrix}};$matrix

${C_{01} = \begin{bmatrix}0 & 0_{r \times k} \\C_{\theta} & 0\end{bmatrix}};$matrix

${D_{01} = \begin{bmatrix}0_{r} & 0 \\0 & D_{\theta\theta}\end{bmatrix}};$matrix

${D_{02} = \begin{bmatrix}0_{r \times p\; 1} \\D_{\theta\; 1}\end{bmatrix}};$matrix C₀₂=[C₁ 0_(p1×k)]; matrix D₀₃=[0_(p1×r) D_(1θ)]; and matrixD₀₄=D₁₁.

Step 4.1: if there is a symmetric positive-definite matrix X, thesymmetrical matrix

$P = \begin{bmatrix}Q & S \\S^{T} & R\end{bmatrix}$making the formula (37) and the formula (38) valid;

$\begin{matrix}{{\begin{bmatrix}I & 0 & 0 \\0 & I & 0 \\0 & 0 & I \\\overset{\_}{A} & {\overset{\_}{B}}_{\theta} & {\overset{\_}{B}}_{w} \\{\overset{\_}{C}}_{\theta} & {\overset{\_}{D}}_{\theta\theta} & {\overset{\_}{D}}_{\theta\; w} \\{\overset{\_}{C}}_{w} & {\overset{\_}{D}}_{w\;\theta} & {\overset{\_}{D}}_{ww}\end{bmatrix}^{T}\begin{bmatrix}0 & 0 & 0 & X & 0 & 0 \\0 & Q & 0 & 0 & S & 0 \\0 & 0 & {{- \gamma}\; I} & 0 & 0 & 0 \\X & 0 & 0 & 0 & 0 & 0 \\0 & S^{T} & 0 & 0 & R & 0 \\0 & 0 & 0 & 0 & 0 & {\frac{1}{\gamma}I}\end{bmatrix}}{\quad{\begin{bmatrix}I & 0 & 0 \\0 & I & 0 \\0 & 0 & I \\\overset{\_}{A} & {\overset{\_}{B}}_{\theta} & {\overset{\_}{B}}_{w} \\{\overset{\_}{C}}_{\theta} & {\overset{\_}{D}}_{\theta\theta} & {\overset{\_}{D}}_{\theta\; w} \\{\overset{\_}{C}}_{w} & {\overset{\_}{D}}_{w\;\theta} & {\overset{\_}{D}}_{ww}\end{bmatrix} < 0}}} & (37) \\{\mspace{79mu}{{\begin{bmatrix}{\Delta(\theta)} & 0 \\0 & {\Delta_{K}(\theta)} \\I & 0 \\0 & I\end{bmatrix}^{T}{P\begin{bmatrix}{\Delta(\theta)} & 0 \\0 & {\Delta_{K}(\theta)} \\I & 0 \\0 & I\end{bmatrix}}} > 0}} & (38)\end{matrix}$

A closed-loop system (36) is asymptotically stable, and the L₂ inducednorm of the closed-loop transfer function from the external input w tothe fault estimation error e_(f) is smaller than a performance indexγ(γ>0). Namely, the solution conditions of the fault estimator K(s,θ)are formula (37) and formula (38),wherein Q, S and R respectivelyrepresent sub-block matrices of P.

Step 4.2: partitioning the symmetric positive-definite matrix X and aninverse matrix X⁻¹ thereof;

$\begin{matrix}{{X = \begin{bmatrix}L & M \\M^{T} & E\end{bmatrix}},{X^{- 1} = \begin{bmatrix}J & N \\N^{T} & F\end{bmatrix}}} & (39)\end{matrix}$

wherein L, M and E respectively represent block matrices of X; J, N andF respectively represent sub-block matrices of X⁻¹.

Because x is the symmetric positive-definite matrix, then

$\begin{matrix}{\begin{bmatrix}L & I \\I & J\end{bmatrix} > 0} & (40)\end{matrix}$

Partitioning the full block scaling matrix P and the inverse matrix{tilde over (P)} thereof

$\begin{matrix}{{P = {\begin{bmatrix}Q & S \\S^{T} & R\end{bmatrix} = \begin{bmatrix}Q_{1} & Q_{2} & S_{1} & S_{2} \\Q_{2}^{T} & Q_{3} & S_{3} & S_{4} \\S_{1}^{T} & S_{3}^{T} & R_{1} & R_{2} \\S_{2}^{T} & S_{4}^{T} & R_{2}^{T} & R_{3}\end{bmatrix}}},{\overset{\sim}{P} = {\begin{bmatrix}\overset{\sim}{Q} & \overset{\sim}{S} \\{\overset{\sim}{S}}^{T} & \overset{\sim}{R}\end{bmatrix} = \begin{bmatrix}{\overset{\sim}{Q}}_{1} & {\overset{\sim}{Q}}_{2} & {\overset{\sim}{S}}_{1} & {\overset{\sim}{S}}_{2} \\{\overset{\sim}{Q}}_{2}^{T} & {\overset{\sim}{Q}}_{3} & {\overset{\sim}{S}}_{3} & {\overset{\sim}{S}}_{4} \\{\overset{\sim}{S}}_{1}^{T} & {\overset{\sim}{S}}_{3}^{T} & {\overset{\sim}{R}}_{1} & {\overset{\sim}{R}}_{2} \\{\overset{\sim}{S}}_{2}^{T} & {\overset{\sim}{S}}_{4}^{T} & {\overset{\sim}{R}}_{2}^{T} & {\overset{\sim}{R}}_{3}\end{bmatrix}}}} & (41)\end{matrix}$

wherein Q₁, Q₂ and Q₃ respectively represent sub-block matrices of Q;S₁, S₂, S₃ and S₄ respectively represent sub-block matrices of S; R₁, R₂and R₃ respectively represent sub-block matrices of R; {tilde over (Q)},{tilde over (S)} and {tilde over (R)} respectively represent sub-blockmatrices of {tilde over (P)}; {tilde over (Q)}₁, {tilde over (Q)}₂ and{tilde over (Q)}₃ respectively represent sub-block matrices of {tildeover (Q)}; {tilde over (S)}₁, {tilde over (S)}₂, {tilde over (S)}₃ and{tilde over (S)}₄ respectively represent sub-block matrices of {tildeover (S)}; {tilde over (R)}₁, {tilde over (R)}₂ and {tilde over (R)}₃respectively represent sub-block matrices of {tilde over (R)}.

Arranging the LMI (37) as

$\begin{matrix}{{\begin{bmatrix}I \\{{U\;\Gamma\; V} + Z}\end{bmatrix}^{T}\begin{bmatrix}0 & 0 & 0 & 0 & 0 & L & M & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & M^{T} & E & 0 & 0 & 0 \\0 & 0 & Q_{1} & Q_{2} & 0 & 0 & 0 & S_{1} & S_{2} & 0 \\0 & 0 & Q_{2}^{T} & Q_{3} & 0 & 0 & 0 & S_{3} & S_{4} & 0 \\0 & 0 & 0 & 0 & {{- \gamma}\; I} & 0 & 0 & 0 & 0 & 0 \\L & M & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\M^{T} & E & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & S_{1}^{T} & S_{3}^{T} & 0 & 0 & 0 & R_{1} & R_{2} & 0 \\0 & 0 & S_{2}^{T} & S_{4}^{T} & 0 & 0 & 0 & R_{2}^{T} & R_{3} & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\frac{1}{\gamma}I}\end{bmatrix}}{\quad{\begin{bmatrix}I \\{{U\;\Gamma\; V} + Z}\end{bmatrix} < 0}}} & (42)\end{matrix}$

wherein matrix

${Z = \begin{bmatrix}A_{0} & B_{01} & B_{02} \\C_{01} & D_{01} & D_{02} \\C_{02} & D_{03} & D_{04}\end{bmatrix}};$matrix

${U = \begin{bmatrix}T_{1} \\T_{5} \\T_{6}\end{bmatrix}};$matrix V=[T₂ T₃ T ₄]; and matrix Γ=Ω.

To satisfy the formula (38), it is required to verify that the formula(38) is valid on all possible tracks of the variable parameter θ, whichis impossible. Therefore, the structure of the full block scaling matrix

$P = \begin{bmatrix}Q & S \\S^{T} & R\end{bmatrix}$is limited to make it valid. For each variable parameter θ, when R≥0,the following formula is valid

$\begin{matrix}{{{\begin{bmatrix}{\theta I} \\I\end{bmatrix}^{T}\begin{bmatrix}Q & S \\S^{T} & R\end{bmatrix}}\begin{bmatrix}{\theta I} \\I\end{bmatrix}} = {{{\theta^{2}Q} + R + {\theta\left( {S^{T} + S} \right)}} \geq {{\theta^{2}\left( {Q + R} \right)} + {\theta\left( {S^{T} + S} \right)}} \geq 0}} & (43)\end{matrix}$

Therefore, Q=−R, and S+S^(T)=0. Namely, the formula (38) is arranged asR>0, Q=−R, S+S ^(T)=0   (44)

To sum up, the solution conditions of the fault estimator K(s,θ) areconverted into formula (40), formula (42) and formula (44).

Step 4.3: arranging the LMI (42) as

$\begin{matrix}{{{V_{\bot}^{T}\begin{bmatrix}I \\Z\end{bmatrix}}^{T}{W\begin{bmatrix}I \\Z\end{bmatrix}}V_{\bot}} < 0} & (45) \\{{{U_{\bot}^{T}\begin{bmatrix}{- Z^{T}} \\I\end{bmatrix}}^{T}{W^{- 1}\begin{bmatrix}{- Z^{T}} \\I\end{bmatrix}}U_{\bot}} > 0} & (46)\end{matrix}$

wherein U_(⊥) and V_(⊥) are respectively the bases of the nuclear spacesof U^(T) and V.

$W = {\begin{bmatrix}0 & 0 & 0 & 0 & 0 & L & M & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & M^{T} & E & 0 & 0 & 0 \\0 & 0 & Q_{1} & Q_{2} & 0 & 0 & 0 & S_{1} & S_{2} & 0 \\0 & 0 & Q_{2}^{T} & Q_{3} & 0 & 0 & 0 & S_{3} & S_{4} & 0 \\0 & 0 & 0 & 0 & {{- \gamma}\; I} & 0 & 0 & 0 & 0 & 0 \\L & M & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\M^{T} & E & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & S_{1}^{T} & S_{3}^{T} & 0 & 0 & 0 & R_{1} & R_{2} & 0 \\0 & 0 & S_{2}^{T} & S_{4}^{T} & 0 & 0 & 0 & R_{2}^{T} & R_{3} & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\frac{1}{\gamma}I}\end{bmatrix}\quad}$

Simplifying the LMIs (45) and (46) through simple calculation as

$\begin{matrix}{{{N_{L}^{T}\begin{bmatrix}I & 0 & 0 \\0 & I & 0 \\0 & 0 & I \\A & B_{\theta} & B_{1} \\C_{\theta} & D_{\theta\theta} & D_{\theta\; 1} \\C_{1} & D_{1\;\theta} & D_{11}\end{bmatrix}}^{T}\begin{bmatrix}0 & 0 & 0 & L & 0 & 0 \\0 & Q_{3} & 0 & 0 & S_{4} & 0 \\0 & 0 & {{- \gamma}\; I} & 0 & 0 & 0 \\L & 0 & 0 & 0 & 0 & 0 \\0 & S_{4}^{T} & 0 & 0 & R_{3} & 0 \\0 & 0 & 0 & 0 & 0 & {\frac{1}{\gamma}I}\end{bmatrix}}{\quad{{\begin{bmatrix}I & 0 & 0 \\0 & I & 0 \\0 & 0 & I \\A & B_{\theta} & B_{1} \\C_{\theta} & D_{\theta\theta} & D_{\theta\; 1} \\C_{1} & D_{1\;\theta} & D_{11}\end{bmatrix}N_{L}} < 0}}} & (47) \\{{{N_{J}^{T}\begin{bmatrix}{- A^{T}} & {- C_{\theta}^{T}} & {- C_{1}^{T}} \\{- B_{\theta}^{T}} & {- D_{\theta\theta}^{T}} & {- D_{1\theta}^{T}} \\{- B_{1}^{T}} & {- D_{\theta 1}^{T}} & {- D_{11}^{T}} \\I & 0 & 0 \\0 & I & 0 \\0 & 0 & I\end{bmatrix}}^{T}\begin{bmatrix}0 & 0 & 0 & J & 0 & 0 \\0 & {\overset{\sim}{Q}}_{3} & 0 & 0 & {\overset{\sim}{S}}_{4} & 0 \\0 & 0 & {{- \frac{1}{\gamma}}I} & 0 & 0 & 0 \\J & 0 & 0 & 0 & 0 & 0 \\0 & {\overset{\sim}{S}}_{4}^{T} & 0 & 0 & {\overset{\sim}{R}}_{3} & 0 \\0 & 0 & 0 & 0 & 0 & {\gamma\; I}\end{bmatrix}}{\quad{{\begin{bmatrix}{- A^{T}} & {- C_{\theta}^{T}} & {- C_{1}^{T}} \\{- B_{\theta}^{T}} & {- D_{\theta\theta}^{T}} & {- D_{1\theta}^{T}} \\{- B_{1}^{T}} & {- D_{\theta 1}^{T}} & {- D_{11}^{T}} \\I & 0 & 0 \\0 & I & 0 \\0 & 0 & I\end{bmatrix}N_{J}} > 0}}} & (48)\end{matrix}$

wherein N_(L) and N_(J) respectively represent the bases of the nuclearspaces of [C₂ D_(2θ) D₂₁] and [B₂ ^(T) D_(θ2) ^(T) D₁₂ ^(T)].

Step 4.4: solving the LMIs (40), (44), (47) and (48) by using an LMItoolkit in MATLAB to obtain an optimal γ value of 0.21 and correspondingmatrix solutions L, J, Q₃, {tilde over (R)}₃, S₄ and {tilde over (S)}₄.

Step 5.1: solving the symmetric positive-definite matrix X, the fullblock scaling matrix P and the inverse matrix {tilde over (P)} thereoffrom the formulas (39) and (41) according to the solved matrix solutionsL, J, Q₃, {tilde over (R)}₃, S₄ and {tilde over (S)}₄.

Step 5.2: according to Schur complement Lemma, expressing the LMI (37)as

$\begin{matrix}{\begin{bmatrix}{{{\overset{\_}{A}}^{T}X} + {X\overset{\_}{A}}} & {{X{\overset{\_}{B}}_{\theta}} + {{\overset{\_}{C}}_{\theta}^{T}S^{T}}} & {X{\overset{\_}{B}}_{w}} & {\overset{\_}{C}}_{\theta}^{T} & {\overset{\_}{C}}_{w}^{T} \\{{{\overset{\_}{B}}_{\theta}^{T}X} + {S{\overset{\_}{C}}_{\theta}}} & {Q + {{\overset{\_}{D}}_{\theta\theta}^{T}S^{T}} + {S{\overset{\_}{D}}_{\theta\theta}}} & {S{\overset{\_}{D}}_{\theta\; w}} & {\overset{\_}{D}}_{\theta\theta}^{T} & {\overset{\_}{D}}_{w\;\theta}^{T} \\{{\overset{\_}{B}}_{w}^{T}X} & {{\overset{\_}{D}}_{\theta\; w}^{T}S^{T}} & {{- \gamma}\; I} & {\overset{\_}{D}}_{\theta\; w}^{T} & {\overset{\_}{D}}_{ww}^{T} \\{\overset{\_}{C}}_{\theta} & {\overset{\_}{D}}_{\theta\theta} & {\overset{\_}{D}}_{\theta\; w} & {- \overset{\sim}{R}} & 0 \\{\overset{\_}{C}}_{w} & {\overset{\_}{D}}_{w\;\theta} & {\overset{\_}{D}}_{ww} & 0 & {{- \gamma}\; I}\end{bmatrix} < 0} & (49)\end{matrix}$

Substituting the value in the closed-loop system (36) to obtainΨ+P ^(T)Ω^(T) Q _(x)+Q _(X) ^(T)ΩP<0   (50)

wherein

${\Psi = \begin{bmatrix}{{A_{0}^{T}X} + {XA}_{0}} & {{XB}_{01} + {C_{01}^{T}S^{T}}} & {XB}_{02} & C_{01}^{T} & C_{02}^{T} \\{{B_{01}^{T}X} + {SC}_{01}} & {Q + {D_{01}^{T}S^{T}} + {SD}_{01}} & {SD}_{02} & D_{01}^{T} & D_{03}^{T} \\{B_{02}^{T}X} & {D_{02}^{T}S^{T}} & {{- \gamma}\; I} & D_{02}^{T} & D_{04}^{T} \\C_{01} & D_{01} & D_{02} & {- \overset{\sim}{R}} & 0 \\C_{02} & D_{03} & D_{04} & 0 & {{- \gamma}\; I}\end{bmatrix}},$P=[T₂ T₃ T₄ 0 0] and Q _(x)=[T₁ ^(T)X T₅ ^(T)S^(T) 0 T₅ ^(T) T₆ ^(T)].

Solving the LMI (50) to obtain a fault estimator matrix Ω.

Step 5.3: obtaining a state space matrix of the fault estimator K(s,θ)

$\begin{matrix}{{{{\begin{bmatrix}{A_{K}(\theta)} & {B_{K}(\theta)} \\{C_{K}(\theta)} & {D_{K}(\theta)}\end{bmatrix} =}\quad}\begin{bmatrix}A_{K} & B_{K\; 1} \\C_{K\; 1} & D_{K\; 1}\end{bmatrix}} + {\left\lbrack \begin{matrix}B_{K\;\theta} \\D_{K\; 1\theta}\end{matrix} \right\rbrack{\Delta_{K}(\theta)}{\left( {I - {D_{K\;{\theta\theta}}{\Delta_{K}(\theta)}}} \right)^{- 1}\begin{bmatrix}C_{K\;\theta} & D_{K\;\theta\; 1}\end{bmatrix}}}} & (51)\end{matrix}$Simulation results at the operating points H=0 km,Ma=0,n₂=90% are shownin FIG. 6(a), FIG. 6(b), FIG. 7(a), FIG. 7(b), FIG. 8(a) and FIG. 8(b),and are compared with the standard H_(∞) method. The simulation resultsshow that the fixed parameter fault estimator designed by the standardH_(∞) method cannot well cope with the change of the variableparameters. The LPV fault estimator designed by the present inventioncan rapidly detect the fault in the system and accurately reconstructthe fault signal, and has obvious performance advantages.

We claim:
 1. A method for fault diagnosis of the sensor and actuator ofan aero-engine based on LFT, comprising the following steps: step 1:establishing an aero-engine state space model using a combination of asmall perturbation method and a linear fitting method; step 1.1:inputting fuel pressure p_(f) under steady operating points into anaero-engine; inputting a fuel pressure step signal U_(pf1) with anamplitude of 0.01 p_(f) into the aero-engine after relative conversionspeed n_(h) of a high pressure rotor of the aero-engine reaches acorresponding steady state; and respectively collecting relativeconversion speed response Y_(nh1) of the high pressure rotor andrelative conversion speed response Y_(nl1) of a low pressure rotoroutputted by the aero-engine; step 1.2: repeating the process of thestep 1 for N times; and respectively collecting the relative conversionspeed response Y_(nhi) of the high pressure rotor and the relativeconversion speed response Y_(nli) of the low pressure rotor under givenfuel pressure p_(fi), wherein i=1,2,3, . . . ,N; step 1.3: constructinga discrete small perturbation state space model of the aero-engine underthe steady operating points according to the linear fitting method bytaking the fuel pressure step signal U_(pfi) as an input variable andtaking the relative conversion speed response Y_(nhi) of the highpressure rotor and the relative conversion speed response Y_(nli) of thelow pressure rotor as state variables; step 1.4: converting the discretesmall perturbation state space model of the aero-engine under the steadyoperating points into a continuous small perturbation state space modelaccording to a sampling period T to obtain the state space model of theaero-engine; $\begin{matrix}\left\{ \begin{matrix}{{\overset{.}{x}}_{p} = {{A_{pi}x_{p}} + {B_{pi}u}}} \\{y_{p} = {{C_{pi}x_{p}} + {D_{pi}u}}}\end{matrix} \right. & (1)\end{matrix}$ wherein the state variable is x_(p)=[Y_(nl) ^(T) Y_(nh)^(T)]^(T)∈R^(n); {dot over (x)}_(p) represents a first derivative ofx_(p); an input variable is u=U_(pf)∈R^(t); an output variable isy_(p)=Y_(nh)∈R^(m); A_(pi), B_(pi), C_(pi) and D_(pi) are system statespace matrices; C_(pi)=C_(p)=[0 1]; D_(pi)=D_(p)=0; R^(n), R^(t) andR^(m) respectively represent sets of real numbers with dimensions of n,t and m; T represents transposing for the matrices; step 2: establishingan affine parameter-dependent linear-parameter-varying LPV model of theaero-engine; step 2.1: setting the relative conversion speed n_(hi) ofthe high pressure rotor of the aero-engine as a scheduling parameterθ(i), i=1,2,3, . . . ,N; step 2.2: expressing a system matrix A_(p)(θ)and a control matrix B_(p)(θ) of the continuous small perturbation statespace model of the aero-engine as affine parameter-dependent forms, asfollows:A _(p)(θ)=A ₀ +θA ₁, B _(p)(θ)=B ₀ +θB ₁   (2) wherein A₀, A₁, B₀ and B₁respectively represent coefficient matrices to be solved; rewriting theformula (2) into $\begin{matrix}{{{A_{p}(\theta)} = {\left\lbrack {I\mspace{14mu}\theta\; I} \right\rbrack\begin{bmatrix}A_{0} \\A_{1}\end{bmatrix}}},{{B_{p}(\theta)} = {\left\lbrack {I\mspace{14mu}\theta\; I} \right\rbrack\begin{bmatrix}B_{0} \\B_{1}\end{bmatrix}}}} & (3)\end{matrix}$ wherein I is a unit matrix; then $\begin{matrix}{{\begin{bmatrix}A_{0} \\A_{1}\end{bmatrix} = {\left\lbrack {I\mspace{20mu}\theta\; I} \right\rbrack^{+}{A_{p}(\theta)}}},{\begin{bmatrix}B_{0} \\B_{1}\end{bmatrix} = {\left\lbrack {I\mspace{20mu}\theta\; I} \right\rbrack^{+}{B_{p}(\theta)}}}} & (4)\end{matrix}$ wherein [I θI]⁺ is Moore-Penrose pseudo- inverse of [IθI], i.e., the system matrix A_(p)(θ) and the control matrix B_(p)(θ) ofthe solved affine parameter-dependent LPV model of the aero-engine; step2.3: establishing the affine parameter-dependent LPV model of theaero-engine{dot over (x)} _(p) =A _(p)(θ)x _(p) +B _(p)(θ)uy _(p) =C _(p) x _(p) +D _(p) u   (5); step 3: converting the affineparameter-dependent LPV model of the aero-engine with perturbation andsensor and actuator fault into a linear fractional transformation (LFT)structure, and establishing an H_(∞) synthesis framework of an LPV faultestimator of the aero-engine; step 3.1: expressing the affineparameter-dependent LPV model P(s,θ) of the aero-engine havingperturbation and sensor and actuator fault into{dot over (x)} _(p) =A _(p)(θ)x _(p) +B _(p)(θ)u+E _(p) d+F _(p) fy _(p) =C _(p) x _(p) +D _(p) u+G _(p) d+H _(p) f   (6) wherein d∈R^(q)is a perturbation signal; f∈R^(l) is a fault signal comprising sensorfault and actuator fault; R^(q) and R^(l) respectively represent sets ofreal numbers with dimensions of q and l; E_(p), F_(p), G_(p) and H_(p)are system state space matrices; an upper LFT structure of P(s,θ) isexpressed into $\begin{matrix}\left\{ \begin{matrix}{{\begin{bmatrix}{\overset{.}{x}}_{p} \\z_{\theta} \\y_{p}\end{bmatrix} = {\begin{bmatrix}A_{p} & B_{p\;\theta} & B_{pw} \\C_{p\;\theta} & D_{p\;{\theta\theta}} & D_{p\;\theta\; w} \\C_{pw} & D_{{pw}\;\theta} & D_{pww}\end{bmatrix}\begin{bmatrix}x_{p} \\w_{\theta} \\w\end{bmatrix}}},{w = \begin{bmatrix}u \\d \\f\end{bmatrix}}} \\{w_{\theta} = {{\Delta(\theta)}z_{\theta}}}\end{matrix} \right. & (7)\end{matrix}$ wherein an external input variable is w=[u^(T) d^(T)f^(T)]^(T)∈R^(p1); w_(θ)∈R^(r) is an output variable of a time varyingpart Δ(θ)=θI; z_(θ)∈R^(r) is an input variable of the time varying partΔ(θ)=θI; A_(p), B_(pθ), B_(pw), C_(pθ), C_(pw), D_(pθθ), D_(pθw),D_(pwθ) and D_(pww) are system state space matrices; R^(p1) and R^(r)respectively represent sets of real numbers with dimensions of p1 and r;p1=t+q+l, i.e., the dimension p1 of the external input variable w isequal to the sum of the dimension t of the input variable u of theaero-engine, the dimension q of the perturbation signal d and thedimension l of the fault signal f; step 3.2: setting the form of thefault estimator K(s,θ) as follows $\begin{matrix}\left\{ \begin{matrix}{{\overset{.}{x}}_{K} = {{{A_{K}(\theta)}x_{K}} + {{B_{K}(\theta)}u_{K}}}} \\{\hat{f} = {{{C_{K}(\theta)}x_{K}} + {{D_{K}(\theta)}u_{K}}}}\end{matrix} \right. & (8)\end{matrix}$ wherein x_(K)∈R^(k) is a state variable of the faultestimator K(s,θ); {dot over (x)}_(K) represents a first derivative ofx_(K); R ^(k) represents a set of real numbers with a dimension of k;u_(K)=[u^(T) y_(p) ^(T)]^(T)∈R^(p2) is an input variable of K(s,θ);p2=t+m, i.e., the dimension p2 of the input variable u_(K) of K(s,θ) isequal to the sum of the dimension t of the input variable u of theaero-engine and the dimension m of the output variable y_(p) of theaero-engine; {circumflex over (f)}∈R^(l) is an output variable ofK(s,θ), i.e., an estimated value of the fault signal f; A_(K)(θ),B_(K)(θ), C_(K)(θ) and D_(K)(θ) are system state space matrices; K(s,θ)is express into a lower LFT structure as follows: $\begin{matrix}\left\{ \begin{matrix}{{\begin{bmatrix}{\overset{.}{x}}_{K} \\\hat{f} \\z_{K}\end{bmatrix} = {\begin{bmatrix}A_{K} & B_{K\; 1} & B_{K\;\theta} \\C_{K\; 1} & D_{K\; 11} & D_{K\; 1\theta} \\C_{K\;\theta} & D_{K\;{\theta 1}} & D_{K\;{\theta\theta}}\end{bmatrix}\begin{bmatrix}x_{K} \\u_{K} \\w_{K}\end{bmatrix}}},{u_{K} = \begin{bmatrix}u \\y_{p}\end{bmatrix}}} \\{w_{K} = {{\Delta_{K}(\theta)}z_{K}}}\end{matrix} \right. & (9)\end{matrix}$ wherein w_(K)∈R^(r) is an output variable of the timevarying part Δ_(K)(θ)=θI; z_(K)∈R^(r) is an input variable of the timevarying part Δ_(K)(θ)=A_(K), B_(K1), B_(Kθ), C_(K1), C_(Kθ), D_(K11),D_(K1θ), D_(Kθ1) and D_(Kθθ) are system state space matrices; step 3.3:according to the time varying part Δ(θ) in the LPV model P(s,θ) of theaero-engine and the time varying part Δ_(K)(θ) in the fault estimatorK(s,θ), expressing the H_(∞) synthesis framework of the LPV faultestimator as: $\begin{matrix}{\begin{bmatrix}\begin{matrix}{\overset{.}{x}}_{p} \\\frac{x_{K}}{z_{K}}\end{matrix} \\\frac{z_{\theta}}{e_{f}}\end{bmatrix} = {\begin{bmatrix}\overset{\_}{A} & {\overset{\_}{B}}_{\theta} & {\overset{\_}{B}}_{w} \\{\overset{\_}{C}}_{\theta} & {\overset{\_}{D}}_{\theta\theta} & {\overset{\_}{D}}_{\theta\; w} \\{\overset{\_}{C}}_{w} & {\overset{\_}{D}}_{w\;\theta} & {\overset{\_}{D}}_{ww}\end{bmatrix}\begin{bmatrix}\begin{matrix}x_{p} \\\frac{x_{K}}{w_{K}}\end{matrix} \\\frac{w_{\theta}}{w}\end{bmatrix}}} & (10)\end{matrix}$ wherein e_(f)={circumflex over (f)}−f is a faultestimation error; system matrix Ā=A₀+T₁ΩT₂; system matrix B_(θ)=B₀₁+T₁ΩT₃; system matrix B _(w)=B₀₂+T₁ΩT₄; system matrix C_(θ)=C₀₁+T₅ΩT₂; system matrix D _(θθ)=D₀₁+T₅ΩT₃; system matrix D_(θw)=D₀₂+T₅ΩT₄; system matrix C _(w)=C₀₂+T₆ΩT₂; system matrix D_(wθ)=D₀₃+T₆ΩT₃; system matrix D _(ww)=D₀₄+T₆ΩT₄; fault estimator matrix${\Omega = \begin{bmatrix}A_{K} & B_{K\; 1} & B_{K\;\theta} \\C_{K\; 1} & D_{K\; 11} & D_{K\; 1\theta} \\C_{K\;\theta} & D_{K\;{\theta 1}} & D_{K\;{\theta\theta}}\end{bmatrix}};$ matrix ${T_{1} = \begin{bmatrix}0 & B_{2} & 0_{n \times r} \\I_{k} & 0 & 0\end{bmatrix}};$ matrix ${T_{2} = \begin{bmatrix}0 & I_{k} \\C_{2} & 0 \\0_{r \times n} & 0\end{bmatrix}};$ matrix ${T_{3} = \begin{bmatrix}0_{k \times r} & 0 \\0 & D_{2\theta} \\I_{r} & 0\end{bmatrix}};$ matrix ${T_{4} = \begin{bmatrix}0_{k \times p\; 1} \\D_{21} \\0_{r \times p\; 1}\end{bmatrix}};$ matrix ${T_{5} = \begin{bmatrix}0_{r \times k} & 0 & I_{r} \\0 & D_{\theta\; 2} & 0\end{bmatrix}};$ matrix T₆=[0_(p1×k) D₁₂ 0_(p1×r)]; matrix${A_{0} = \begin{bmatrix}A & 0 \\0 & 0_{k}\end{bmatrix}};$ matrix ${B_{0\; 1} = \begin{bmatrix}0 & B_{\theta} \\0_{k \times r} & 0\end{bmatrix}};$ matrix ${B_{02} = \begin{bmatrix}B_{1} \\0_{k \times p\; 1}\end{bmatrix}};$ matrix ${C_{0\; 1} = \begin{bmatrix}0 & 0_{r \times k} \\C_{\theta} & 0\end{bmatrix}};$ matrix ${D_{01} = \begin{bmatrix}0_{r} & 0 \\0 & D_{\theta\theta}\end{bmatrix}};$ matrix ${D_{02} = \begin{bmatrix}0_{r \times p\; 1} \\D_{\theta\; 1}\end{bmatrix}};$ matrix C₀₂=[C₁ 0_(p1×k)]; matrix D₀₃=[0_(p1×r) D_(1θ)];matrix D₀₄=D₁₁; matrix A=A_(p); matrix B_(θ)=B_(pθ); matrix B₁=B_(pw);matrix B₂=0_(n×l); matrix C_(θ)=C_(pθ); matrix D_(θθ)=D_(pθθ); matrixD_(θ1)=D_(pθw); matrix D_(θ2)=0_(r×l); matrix C₁=0_(p1×n); matrixD_(1θ)=0_(p1×r); matrix ${D_{11} = \begin{bmatrix}0_{l \times {({{p\; 1} - l})}} & {- I_{l}} \\0_{{({{p\; 1} - l})} \times {({{p\; 1} - l})}} & 0_{{({{p\; 1} - l})} \times l}\end{bmatrix}};$ matrix ${D_{12} = \begin{bmatrix}I_{l} \\0_{{({{p\; 1} - l})} \times l}\end{bmatrix}};$ matrix ${C_{2} = \begin{bmatrix}0_{t \times n} \\C_{pw}\end{bmatrix}};$ matrix ${D_{2\theta} = \begin{bmatrix}0_{t \times r} \\C_{{pw}\;\theta}\end{bmatrix}};$ matrix ${D_{21} = \begin{bmatrix}I_{t} & 0_{t \times q} & 0_{t \times l} \\\; & D_{pww} & \;\end{bmatrix}};$ D₂₂=0_(p2×l); n represents the dimension of the statevariable x_(p) of the aero-engine; r represents the dimension of theoutput variable w_(θ) of the time varying part Δ(θ) and the outputvariable w_(K) of the time varying part Δ_(K)(θ); k represents thedimension of the state variable x_(K) of the fault estimator K(s,θ);step 4: solving a set of linear matrix inequalities (LMIs) to obtain thesolution conditions of the fault estimator; step 4.1: obtaining thesolution conditions of the fault estimator K(s,θ), i.e., $\begin{matrix}{{\begin{bmatrix}I & 0 & 0 \\0 & I & 0 \\0 & 0 & I \\\overset{\_}{A} & {\overset{\_}{B}}_{\theta} & {\overset{\_}{B}}_{w} \\{\overset{\_}{C}}_{\theta} & {\overset{\_}{D}}_{\theta\theta} & {\overset{\_}{D}}_{\theta\; w} \\{\overset{\_}{C}}_{w} & {\overset{\_}{D}}_{w\;\theta} & {\overset{\_}{D}}_{ww}\end{bmatrix}^{T}\begin{bmatrix}0 & 0 & 0 & X & 0 & 0 \\0 & Q & 0 & 0 & S & 0 \\0 & 0 & {{- \gamma}\; I} & 0 & 0 & 0 \\X & 0 & 0 & 0 & 0 & 0 \\0 & S^{T} & 0 & 0 & R & 0 \\0 & 0 & 0 & 0 & 0 & {\frac{1}{\gamma}I}\end{bmatrix}}{\quad{\begin{bmatrix}I & 0 & 0 \\0 & I & 0 \\0 & 0 & I \\\overset{\_}{A} & {\overset{\_}{B}}_{\theta} & {\overset{\_}{B}}_{w} \\{\overset{\_}{C}}_{\theta} & {\overset{\_}{D}}_{\theta\theta} & {\overset{\_}{D}}_{\theta\; w} \\{\overset{\_}{C}}_{w} & {\overset{\_}{D}}_{w\;\theta} & {\overset{\_}{D}}_{ww}\end{bmatrix} < 0}}} & (11) \\{\mspace{79mu}{{\begin{bmatrix}{\Delta(\theta)} & 0 \\0 & {\Delta_{K}(\theta)} \\I & 0 \\0 & I\end{bmatrix}^{T}{P\begin{bmatrix}{\Delta(\theta)} & 0 \\0 & {\Delta_{K}(\theta)} \\I & 0 \\0 & I\end{bmatrix}}} > 0}} & (12)\end{matrix}$ wherein X is a symmetric positive-definite matrix; a fullblock scaling matrix $P = \begin{bmatrix}Q & S \\S^{T} & R\end{bmatrix}$ is a symmetric matrix; γ>0 is a performance index; Q, Sand R respectively represent sub-block matrices of P; step 4.2:partitioning the symmetric positive-definite matrix X and an inversematrix X⁻¹ thereof; $\begin{matrix}{{X = \begin{bmatrix}L & M \\M^{T} & E\end{bmatrix}},{X^{- 1} = \begin{bmatrix}J & N \\N^{T} & F\end{bmatrix}}} & (13)\end{matrix}$ wherein L, M and E respectively represent block matricesof X; J, N and F respectively represent sub-block matrices of X⁻¹;partitioning the full block scaling matrix P and the inverse matrix{tilde over (P)} thereof $\begin{matrix}{{P = {\begin{bmatrix}Q & S \\S^{T} & R\end{bmatrix} = \left\lbrack \begin{matrix}\begin{matrix}Q_{1} & Q_{2} \\Q_{2}^{T} & Q_{3}\end{matrix} & \begin{matrix}S_{1} & S_{2} \\S_{3} & S_{4}\end{matrix} \\\begin{matrix}S_{1}^{T} & S_{3}^{T} \\S_{2}^{T} & S_{4}^{T}\end{matrix} & \begin{matrix}R_{1} & R_{2} \\R_{2}^{T} & R_{3}\end{matrix}\end{matrix} \right\rbrack}},{\overset{\sim}{P} = {\begin{bmatrix}\overset{\sim}{Q} & \overset{\sim}{S} \\{\overset{\sim}{S}}^{T} & \overset{\sim}{R}\end{bmatrix} = \left\lbrack \begin{matrix}\begin{matrix}{\overset{\sim}{Q}}_{1} & {\overset{\sim}{Q}}_{2} \\{\overset{\sim}{Q}}_{2}^{T} & {\overset{\sim}{Q}}_{3}\end{matrix} & \begin{matrix}{\overset{\sim}{S}}_{1} & {\overset{\sim}{S}}_{2} \\{\overset{\sim}{S}}_{3} & {\overset{\sim}{S}}_{4}\end{matrix} \\\begin{matrix}{\overset{\sim}{S}}_{1}^{T} & {\overset{\sim}{S}}_{3}^{T} \\{\overset{\sim}{S}}_{2}^{T} & {\overset{\sim}{S}}_{4}^{T}\end{matrix} & \begin{matrix}{\overset{\sim}{R}}_{1} & {\overset{\sim}{R}}_{2} \\{\overset{\sim}{R}}_{2}^{T} & {\overset{\sim}{R}}_{3}\end{matrix}\end{matrix} \right\rbrack}}} & (14)\end{matrix}$ wherein Q₁, Q₂ and Q₃ respectively represent sub-blockmatrices of Q; S₁, S₂, S₃ and S₄ respectively represent sub-blockmatrices of S; R₁, R₂ and R₃ respectively represent sub-block matricesof R; {tilde over (Q)}, {tilde over (S)} and {tilde over (R)}respectively represent sub-block matrices of {tilde over (P)}; {tildeover (Q)}₁, {tilde over (Q)}₂ and {tilde over (Q)}₃ respectivelyrepresent sub-block matrices of {tilde over (Q)}; {tilde over (S)}₁,{tilde over (S)}₂, {tilde over (S)}₃ and {tilde over (S)}₄ respectivelyrepresent sub-block matrices of {tilde over (S)}; {tilde over (R)}₁,{tilde over (R)}₂ and {tilde over (R)}₃ respectively represent sub-blockmatrices of {tilde over (R)}; simplifying the solution conditions of thefault estimator K(s,θ), i.e., $\begin{matrix}{{{N_{L}^{T}\begin{bmatrix}I & 0 & 0 \\0 & I & 0 \\0 & 0 & I \\A & B_{\theta} & B_{1} \\C_{\theta} & D_{\theta\theta} & D_{\theta\; 1} \\C_{1} & D_{1\;\theta} & D_{11}\end{bmatrix}}^{T}\begin{bmatrix}0 & 0 & 0 & L & 0 & 0 \\0 & Q_{3} & 0 & 0 & S_{4} & 0 \\0 & 0 & {{- \gamma}\; I} & 0 & 0 & 0 \\L & 0 & 0 & 0 & 0 & 0 \\0 & S_{4}^{T} & 0 & 0 & R_{3} & 0 \\0 & 0 & 0 & 0 & 0 & {\frac{1}{\gamma}I}\end{bmatrix}}{\quad{{\begin{bmatrix}I & 0 & 0 \\0 & I & 0 \\0 & 0 & I \\A & B_{\theta} & B_{1} \\C_{\theta} & D_{\theta\theta} & D_{\theta\; 1} \\C_{1} & D_{1\;\theta} & D_{11}\end{bmatrix}N_{L}} < 0}}} & (15) \\{{{N_{J}^{T}\begin{bmatrix}{- A^{T}} & {- C_{\theta}^{T}} & {- C_{1}^{T}} \\{- B_{\theta}^{T}} & {- D_{\theta\theta}^{T}} & {- D_{1\;\theta}^{T}} \\{- B_{1}^{T}} & {- D_{\theta\; 1}^{T}} & {- D_{11}^{T}} \\I & 0 & 0 \\0 & I & 0 \\0 & 0 & I\end{bmatrix}}^{T}\begin{bmatrix}0 & 0 & 0 & J & 0 & 0 \\0 & {\overset{\sim}{Q}}_{3} & 0 & 0 & {\overset{\sim}{S}}_{4} & 0 \\0 & 0 & {{- \frac{1}{\gamma}}\; I} & 0 & 0 & 0 \\J & 0 & 0 & 0 & 0 & 0 \\0 & {\overset{\sim}{S}}_{4}^{T} & 0 & 0 & {\overset{\sim}{R}}_{3} & 0 \\0 & 0 & 0 & 0 & 0 & {\gamma\; I}\end{bmatrix}}{\quad{{\begin{bmatrix}{- A^{T}} & {- C_{\theta}^{T}} & {- C_{1}^{T}} \\{- B_{\theta}^{T}} & {- D_{\theta\theta}^{T}} & {- D_{1\;\theta}^{T}} \\{- B_{1}^{T}} & {- D_{\theta\; 1}^{T}} & {- D_{11}^{T}} \\I & 0 & 0 \\0 & I & 0 \\0 & 0 & I\end{bmatrix}N_{J}} > 0}}} & (16)\end{matrix}$ $\begin{matrix}\begin{bmatrix}L & I \\I & J\end{bmatrix} & (17) \\{{R > 0},{Q = {- R}},{{S + S^{T}} = 0}} & (18)\end{matrix}$ wherein N_(L) and N_(J) respectively represent the basesof the nuclear spaces of [C₂ D_(2θ) D₂₁] and [B₂ ^(T) D_(θ2) ^(T) D₁₂^(T)]; step 4.3: solving the LMIs (15)-(18) to obtain matrix solutionsL, J, Q₃, {tilde over (R)}₃, S₄ and {tilde over (S)}₄; step 5: designingthe fault estimator in combination with the LFT structure to realizefault diagnosis of the sensor and actuator of the aero-engine; step 5.1:solving the symmetric positive-definite matrix X, the full block scalingmatrix P and the inverse matrix {tilde over (P)} thereof from theformulas (13) and (14) according to the solved matrix solutions L, J,Q₃, {tilde over (R)}₃, S₄ and {tilde over (S)}₄; step 5.2: according toSchur complement Lemma, expressing the LMI (11) as $\begin{matrix}{\begin{bmatrix}{{{\overset{\_}{A}}^{T}X} + {X\;\overset{\_}{A}}} & {{X\;{\overset{\_}{B}}_{\theta}} + {{\overset{\_}{C}}_{\theta}^{T}S^{T}}} & {X\;{\overset{\_}{B}}_{w}} & {\overset{\_}{C}}_{\theta}^{T} & {\overset{\_}{C}}_{w}^{T} \\{{{\overset{\_}{B}}_{\theta}^{T}X} + {S\;{\overset{\_}{C}}_{\theta}}} & {Q + {{\overset{\_}{D}}_{\theta\theta}^{T}S^{T}} + {S\;{\overset{\_}{D}}_{\theta\theta}}} & {S\;{\overset{\_}{D}}_{\theta\; w}} & {\overset{\_}{D}}_{\theta\theta}^{T} & {\overset{\_}{D}}_{\theta\; w}^{T} \\{{\overset{\_}{B}}_{w}^{T}X} & {{\overset{\_}{D}}_{\theta\; w}^{T}S^{T}} & {{- \gamma}\; I} & {\overset{\_}{D}}_{\theta\; w}^{T} & {\overset{\_}{D}}_{w\; w}^{T} \\{\overset{\_}{C}}_{\theta} & {\overset{\_}{D}}_{\theta\theta} & {\overset{\_}{D}}_{\theta\; w} & {- R} & 0 \\{\overset{\_}{C}}_{w} & {\overset{\_}{D}}_{w\;\theta} & {\overset{\_}{D}}_{ww} & 0 & {{- \gamma}\; I}\end{bmatrix} < 0} & (19)\end{matrix}$ solving the LMI (19) to obtain a fault estimator matrix Ω;step 5.3: obtaining a state space matrix of the fault estimator K(s,θ)$\begin{matrix}{\begin{bmatrix}{A_{K}(\theta)} & {B_{K}(\theta)} \\{C_{K}(\theta)} & {D_{K}(\theta)}\end{bmatrix} = {\quad{\begin{bmatrix}A_{K} & B_{K\; 1} \\C_{K\; 1} & D_{K\; 11}\end{bmatrix} + {\begin{bmatrix}B_{K\;\theta} \\D_{K\; 1\;\theta}\end{bmatrix}{\Delta_{K}(\theta)}{{\left( {I - {D_{K\;\theta\;\theta}{\Delta_{K}(\theta)}}} \right)^{- 1}\left\lbrack {C_{K\;\theta}\mspace{20mu} D_{K\;\theta\; I}} \right\rbrack}.}}}}} & (20)\end{matrix}$